Question

Let } X \text { be a Banach space. Assume that } x_{n} \rightarrow x \text { and } f_{n} \in X^{*} \text { are }\\\ &\text { such that } f_{n} \stackrel{w^{*}}{\rightarrow} f \text { . Show that } f_{n}\left(x_{n}\right) \rightarrow f(x) \end{aligned} $$

Short answer

Yes, \( f_n(x_n) \rightarrow f(x) \)

Step-by-step solution

- Understand the Problem

The problem involves a Banach space X, a sequence of elements \( x_n \) converging to \( x \), and a sequence of continuous linear functionals \( f_n \) converging weakly-* to \( f \). The goal is to show that \( f_n(x_n) \) converges to \( f(x) \).

- Recall Definitions

Recall that \( f_n \) converges weakly-* to \( f \) means for all \( y \in X \), \( f_n(y) \to f(y) \). Also, know that \( f_n \) are elements of \( X^* \), the dual space of X.

- Use the Uniform Boundedness Principle

Apply the Uniform Boundedness Principle which states that the sequence \( f_n \) is uniformly bounded. Therefore, there exists a constant M such that \( \forall n, \|f_n\| \leq M \).

- Apply Pointwise Convergence

Use the fact that \( f_n \stackrel{w^{*}}{\rightarrow} f \), then \( f_n(y) \to f(y) \) for all \( y \in X \), and take \( y = x \) to get \( f_n(x) \to f(x) \).

- Combine Results Using Triangle Inequality

Show that \( f_n(x_n) \rightarrow f(x) \) by combining the previous steps. Use the triangle inequality to handle \( f_n(x_n) - f(x) \) as follows: \[ |f_n(x_n) - f(x)| \leq |f_n(x_n) - f_n(x)| + |f_n(x) - f(x)|.\]

- Apply Limiting Arguments

Note that since \( x_n \rightarrow x \), \( |f_n(x_n) - f_n(x)| \rightarrow 0 \) due to the continuity of \( f_n \). Since \( f_n \stackrel{w^{*}}{\rightarrow} f \), \( |f_n(x) - f(x)| \rightarrow 0 \). Combine these to get the desired result.

- Conclude the Proof

Summarize that combining the two limiting results from Step 6 gives us \[ \textstyle\boxed{ f_n(x_n) \rightarrow f(x) }\]

Key concepts

weak-* convergence

Weak-* convergence is an important concept in functional analysis, especially within Banach spaces. If a sequence of functionals \({f_n}\) in the dual space \({X^*}\) converges weakly-* to a functional \({f}\), it means that for every element \({y}\) in the Banach space \({X}\), the sequence \({f_n(y)}\) converges to \({f(y)}\). This type of convergence is weaker than norm convergence but still provides valuable information about the behavior of the functionals. It is essential in scenarios like the one presented in the exercise, where we want to show that \({f_n(x_n) \rightarrow f(x)}\) given certain conditions.

continuous linear functional

A continuous linear functional is a linear map from a Banach space \({X}\) to the field of scalars (typically \(\text{R}\) or \(\text{C}\)). The continuity of these functionals ensures that small changes in the input lead to small changes in the output. This property is crucial because it guarantees that the functional responds predictably to convergence processes within \({X}\). In the context of this exercise, \({f_n}\) and \({f}\) are continuous linear functionals, allowing us to apply various convergence theorems and principles to achieve the desired result.

Uniform Boundedness Principle

The Uniform Boundedness Principle is a cornerstone of functional analysis. It states that if a sequence of continuous linear functionals in a Banach space is pointwise bounded, then it is uniformly bounded. In mathematical terms, if \({f_n(y)}\) is bounded for each \({y \in X}\), then there exists a constant \({M}\) such that \({\forall n, \|f_n\| \leq M}\). This principle is applied in Step 3 of the solution to ensure that we have a uniform bound on the \({f_n}\), facilitating the subsequent steps to show \({f_n(x_n) \rightarrow f(x)}\).

pointwise convergence

Pointwise convergence refers to a situation where a sequence of functions \({f_n}\) converges to a function \({f}\) at every point \({x}\) in the domain. For every \({x}\) in \({X}\), \({f_n(x) \rightarrow f(x)}\). This type of convergence is useful in the given problem for handling cases where both the input sequence \({x_n}\) converges to \({x}\) and the functionals \({f_n}\) converge weakly-* to \({f}\). With pointwise convergence, we deduce that \({f_n(x) \rightarrow f(x)}\) for any fixed \({x}\).

triangle inequality

The triangle inequality is a fundamental property in normed vector spaces, including Banach spaces. It states that for any vectors \({a}\) and \({b}\), \({\|a + b\textbar \textbar \leq \textbar \textbar a\textbar \textbar + \textbar \textbar b\textbar \textbar \). This inequality is used in Step 5 to manage the difference between \({f_n(x_n)}\) and \({f(x)}\). Specifically, it helps break down the expression \({f_n(x_n) - f(x)}\) into more manageable parts: \[ \textbar f_n(x_n) - f(x) \textbar \leq \textbar f_n(x_n) - f_n(x) \textbar + \textbar f_n(x) - f(x) \textbar. \] By analyzing these individual terms, we can approach the desired convergence step by step.